Press-Schechter Formalism and The PBH Mass Distributions

Primordial black holes (PBHs) can form during radiation domination from rare primordial perturbations that re-enter the Hubble radius and undergo gravitational collapse. We derive PBH mass distributions using Press–Schechter theory completed by the excursion-set first-crossing construction. We define the smoothed density contrast ${\delta }_R$ and its variance $S(R)={\sigma }^2(R)$, and connect $S$ to the primordial curvature spectrum ${\mathcal{P}}_{\mathcal{R}}(k)$ through the radiation-era transfer. For Gaussian statistics and a constant collapse threshold ${\delta }_c$, the formation fraction is an $\mathrm{erfc}$ tail with a controlled rare-event asymptotic. For a sharp-$k$ filter, $\delta (S)$ is Markovian; solving the diffusion equation with an absorbing barrier yields the first-crossing density $f(S)=\frac{{\delta }_c}{\sqrt{2\pi }}{S}^{-3\mathrm{/}2}\exp \text{\hspace{0.17em}⁣}(-{\delta }_c^2\mathrm{/}(2S))$. This gives a differential formation fraction $\mathrm{d}\beta \mathrm{/}\mathrm{d}\ln M=f(S)\mid \mathrm{d}S\mathrm{/}\mathrm{d}\ln M\mid$ and a mass-conserving formation-era mass function $\mathrm{d}{n}_{\mathrm{P}\mathrm{B}\mathrm{H}}\mathrm{/}\mathrm{d}M$. We then map to the present-day PBH dark-matter fraction per logarithmic mass, $f_{\mathrm{P}\mathrm{B}\mathrm{H}}(M)$, using horizon-entry scaling $M\propto k^{-2}$ and radiation-era redshifting.